\section{Introduction}

\subsection{Motivations}

Straight edge detection is a preliminary step of many image analysis
processes. Therefore it is always an active reasearch topic centered
on the quest of still faster, more accurate or more robust-to-noise
methods.

{\it TOWRITE : petit \'etat de l'art en r\'esumant IWCIA'09 et en ajoutant
quelques id\'ees perso (Hough, local, ...).
Parameter-space-based methods : robust to noise, well suited to
supervided context \cite{EvenMalavaud00}.
Most of works aim at reducing their time complexity. }

These methods rarely provide a direct measure of the quality of the output
edge, such as sharpness, connectivity or scattering.
Some information may often be drawn from their specific context, for example
through an analysis of the peak in a Hough transform accumulator, or
TO COMPLETE.
In particular, the accuracy of the edge orientation may be quite critical
in some application contexts, such as computer vision.

In digital geometry, the notion of blurred segment \cite{DebledAl05,Buzer07}
was introduced to cope with the image noise or other sources of
imperfections from the real world. The pre-image of that geometrical object,
ie the space of geometric entities which numerization matches this
blurred segment, may convey useful information to evaluate possible moves in
the 3D interpretations drawn, as a promising extension of former works
on discrete epipolar geometry \cite{NatsumiAl08}.

Our work aims at designing a flexible tool to detect such blurred segment
in gray-level images for as well supervised as unsupervised contexts.
We seek for user-friendly solutions with ideally no parameter to set,
or at least quite few values with intuitive meaning to an end user.

\subsection{Method overview and previous work}

The method we propose is based on a first rough detection in a local area
of the image either defined by the user in supervised context or blindly
explored in automatic mode. The goal is to disclose the presence of an edge.
Therefore, a simple test as the gradient maximal value is performed.

In case of success, refinement steps are run through an exploration of
the image in the direction of the detected edge.
In order to prevent local disturbances, for instance the intersection with
a sharper edge, all the local gradient maxima are successively tested,
and the gradient orientation consistency is checked.

We already designed and experimented an exploratory detector
\cite{KerautretEven09}.

Despite of good performances obtained compared
to other methods from the literature, several drawbacks remained.
At first, a fixed width value was set by the user according to the
application requirements, and detected segments were embedded in that
fixed tolerence whatever their dispersion be. When this dispersion is low,
the blurred segment is free to shiff sidewards, or worst, to rotate, thus
degrading the provided position and rotation measures.

Then two refinement steps were arbitrarily run to cope with most of
the tested data, uselessly when the first one was successfull.
Beyond, there was no guarantee that this could treat all kinds
of data. The search direction is fixed by the detected direction at the
former step, and there is necessarily a limit on this direction
accuracy - at least linked to the restricted directions encoded
in a limited grid - so that other steps would have been necessary
to deal with high resolution images.

Our study relies only on the use of the image gradient, as it provides a
good information on the signal dynamics, and hence the presence of edges.
Trials were made to use also the intensity signal though expensive
correlation techniques, but it was mostly successful for tracking objects
with stable appearance such as metallic pipes \cite{AubryAl17}.

Organisation of the paper : TO WRITE.
